The semigroup algebra , where is a field and a semigroup, is formally defined in the same way as the group algebra . Similarly, a semigroup ring is a variation of the group ring , where the group is replaced by a semigroup . Usually, it is required that have an identity element so that is a unit ring and is a subring of .
The group algebra is the set of all formal expressions
(1)
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where for all and for all but finitely many indices so that for sufficiently large (say, ). Hence, we can write the general element as
(2)
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Assigning
(3)
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defines an isomorphism of -algebras between and the polynomial ring .
More generally, if is the subsemigroup of generated by the elements , for , the semigroup algebra is isomorphic to the subalgebra of the polynomial ring generated by the monomials